An analytical synthesis of fractional order PI^λD^μ controller design

This paper proposed a comprehensive synthesis of fractional order PI^λD^μ (FOPI^λD^μ) controller analytical design, which is illustrated through the typical first order plus normalized time delay (FOPNTD) systems, with fulfilling five frequency-domain specifications simultaneously: phase margin (ϕ_m), gain margin (A_m), phase crossover frequency (ω_pc), gain crossover frequency (ω_gc) and a "flat phase" constraint. The control loop shape can be adjusted with wide freedom according to the five design specifications, which can all be beneficial on optimizing the control system: ω_gc represents the system control bandwidth and response speed, ϕ_m and A_m guarantee the stability, and flat phase constraint keeps the system with iso-damping property on robustness of loop gain variations. The impact of ω_pc adjustment is thoroughly discovered via frequency-domain analysis and also time-domain analysis with low-frequency disturbance and high-frequency noise. The frequency response functions are presented to show the loop-shaping advantages of the proposed synthesis scheme. A further in-depth study on designing guideline is also presented: the feasible region of four specifications, e.g. ω_gc, ω_pc, ϕ_m and A_m, can all be collected and visualized in the multi-dimensional graphics. This feasible region gives users prior information and great flexibility before the controller design. Simulation results using the designed FOPI^λD^μ controller are carried out to demonstrate the performance advantages over the optimized integer-order PID, three-parameter FOPID, fractional filter-fractional order PID and Ziegler-Nichols FOPID controllers.